refinement-ladder
At its defaults it draws iterative refinement from a 24-bit factorisation, κ = 10⁴. A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
refinement-ladder is one function in lib/figures/mixed.js —
mixed precision — refinement, and the threshold at 1/u. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
bits: 11
The arguments are the ones A norm that overflows before it is a norm passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
bits: 24
The arguments are the ones Buying the accuracy back passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
8 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
a double residual recovers the accuracy
and a same-precision residual does not
both runs start from the same solve
LU is for square matrices
matmul shapes agree
past the threshold, refinement does not reach the working precision
reaching what a double solve reaches
the constructed matrix has κ = 10000
Against the rule
It draws a decomposition and prints its residual. It calls
luFactor, refine,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A norm that overflows before it is a norm
The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.
The arithmetic underneathBuying the accuracy back
Factorise in single precision, then correct the answer using residuals computed in double, and the result is what a full double-precision solve would have given. Compute those residuals in single instead and the identical algorithm, at identical cost, recovers nothing.
The arithmetic underneathWhere the hardware went
bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.